Recently, the well-posedness of the nonlinear Schrödinger equation with slowly decaying initial data has attracted attention [1]}, [2]}, in part due to the importance of such initial data for modeling signals. The well-posedness results in this article are proved via a simple contraction mapping argument; similar as in [2]} by the second author. We use Duhamel's formula to write (REF ) as
\(u(t) = \Phi _{f,g}(u) = \cos (t \sqrt{- \Delta }) f + \frac{\sin (t \sqrt{- \Delta }) g}{\sqrt{- \Delta }} \pm \int _0^t \frac{\sin ((t-s) \sqrt{- \Delta })}{\sqrt{- \Delta }} (|u|^{\alpha - 1} u)(s) ds.\) 
